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Theorem reupick3 4319
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick3 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reupick3
StepHypRef Expression
1 df-reu 3376 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 df-rex 3070 . . . . 5 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
3 anass 468 . . . . . 6 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
43exbii 1849 . . . . 5 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
52, 4bitr4i 278 . . . 4 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥((𝑥𝐴𝜑) ∧ 𝜓))
6 eupick 2628 . . . 4 ((∃!𝑥(𝑥𝐴𝜑) ∧ ∃𝑥((𝑥𝐴𝜑) ∧ 𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
71, 5, 6syl2anb 597 . . 3 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
87expd 415 . 2 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓)) → (𝑥𝐴 → (𝜑𝜓)))
983impia 1116 1 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wex 1780  wcel 2105  ∃!weu 2561  wrex 3069  ∃!wreu 3373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-ex 1781  df-nf 1785  df-mo 2533  df-eu 2562  df-rex 3070  df-reu 3376
This theorem is referenced by:  reupick2  4320  fvineqsneq  36597
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